Let \(\alpha \in \text{Wander}_\phi (K)\)  such that there exist non-negative integers \(m>n>0\) , integers \(r\)  and \(s\) , and \(u \in R^*_S\)  such that
\((\phi ^m(\alpha ))^r=u(\phi ^n(\alpha ))^s.\) 
If \(r=0\)  or \(s=0\) , the result follows from Theorem REF , so we may assume that \(rs \ne 0.\)  Also, due to the saturation \(R^*_S\) , we may assume that gcd\((r,s)=1\) . We enlarge \(S\)  by the places for which \(\phi \)  has bad reduction, namely, those \(v \in M_K\)  such that if \(f(z)=c_0+c_1z+...+c_dz^d\)  then \(v(c_i)<0\)  for some \(i\)  or \(v(c_d)>0\) , obtaining a new finite set of places \(S_\phi \) . In fact, proving the results for the larger \(S_f\)  gives results that are stronger than the original statements. In this case, one can see that
\(S_{\phi ^m}\subset S_\phi \text{ for all }m \ge 1.\) 
Since \(rs \ne 0,\)  replacing \(r,s\)  by \(-r,-s\)  if necessary, we may assume that \(r>0\) . We now divide the proof in cases.
Case A: \(\alpha \in R_{S_\phi }\) .
By definition one can check that \(\phi ^k(\alpha ) \in R_{S_\phi }\)  for every \(k\ge 0\) , so that
\(v(\phi ^k(\alpha ))\ge 0 \text{ for all } k \ge 0, v \notin S_\phi .\) 
We now divide this step in some subcases.
Case A.1: \(r>0, s<0\) .
Here, equation REF  becomes
\((\phi ^m(\alpha ))^r(\phi ^n(\alpha ))^t=u \text{ with } t=-s>0. \) 
As \(u \in R^*_{S_\phi },\)  it follows that
\(rv(\phi ^m(\alpha ))+tv((\phi ^n(\alpha ))=0 \text{ for all } v \notin S_\phi .\) 
Since \(r,t>0\) , this implies that
\(v(\phi ^m(\alpha ))=v((\phi ^n(\alpha ))=0 \text{ for all } v \notin S_\phi ,\) 
and hence that \(\phi ^m(\alpha ) \in R^*_S\) . The conclusions come then from Theorem REF .
Case A.2: \(r>0, s\ge 2\) .
Since gcd\((r,s)\) , we can choose \(a\)  and \(b\)  with \(ar+bs=1\)  so that REF  becomes
\(\phi ^m(\alpha )=u^a\left((\phi ^n(\alpha ))^a(\phi ^m(\alpha ))^b \right)^s. \) 
Since \(\phi ^m(\alpha ) \in R_{S_\phi }\)  and \(u \in R^*_{S_\phi }\) , we have that \((\phi ^n(\alpha ))^a(\phi ^m(\alpha ))^b \in R_{S_\phi }\) . If \((\phi ^n(\alpha ))^a(\phi ^m(\alpha ))^b \in R_{S_\phi }^*\) , then \(\phi ^m(\alpha ) \in R_{S_\phi }^*\)  and we have the desired results by Theorem REF . If not, then we also have that \((\phi ^n(\alpha ))^a(\phi ^m(\alpha ))^b \notin k\) . Writing
\(\phi ^m(\alpha )=\phi (\phi ^{m-1}(\alpha )),\) 
we use Lemma REF  to conclude that the exponent \(s\ge 2\)  is effectively bounded in terms of \(\phi \)  and \(K\) .
If we first assume that \(\deg \phi \ge 3\) , then we can apply [1]} to effectively bound the referred \(\phi ^{m-1}(\alpha )\) 's, and then \(m\)  and \(h(\alpha )\)  as desired after using the similar method to the one used in the proof of Theorem REF (b).
If otherwise \(\deg \phi =2\) , we can apply [1]}, and conclude similarly.
Case A.3: \(r\ge 2, s=1\) .
If \(n \ge 2\) , then the same discussion as above holds again (replacing \(m\)  by \(n\)  and \(r\)  by \(s\) ). It is enough then to consider the case \(n=1\) , so that REF  becomes
\(\phi (\alpha )=u^{-1}(\phi ^m(\alpha ))^r.\) 
If \(r \ge 3\) , then we can apply [1]} again to conclude.
If otherwise \(r=2\) , we can apply Theorem REF  to conclude as desired.
Case A.4: \(r=1, s=1\) .
This case is covered by Theorem REF .
Case B: \(\alpha \notin R_{S_\phi }\) .
Choosing \(v \in M_K\setminus S_\phi \)  such that \(v(\alpha )<0\) , we have also that \(v \notin S_{\phi ^k}\)  for all \(k\ge 1\) . In this case, we have that \(v(\phi ^k(\alpha ))=d^kv(\alpha )\)  for all \(k\ge 0\)  and hence REF  implies
\(rd^mv(\alpha )=sd^nv(\alpha ).\) 
Therefore \(r=1\)  and \(s=d^{m-n}\) , and REF  becomes
\(\phi ^{n+k}(\alpha )=u(\phi ^k(\alpha ))^{d^{n}}.\) 
If \(n=1\) , \(\phi ^{1+k}(\alpha )=u(\phi ^k(\alpha ))^{d^{n}},\)  and we can use Theorem REF (a) applied to the rational function \(f(X)/X^d\)  to conclude the desired.
Otherwise, we first enlarge \(S_\phi \)  in a controlled way such that \(\text{disc}(\phi ), \text{disc}(\phi ^2) \in R^*_{S_\phi }\) . Moreover, we note that the group \(R^*_{S_\phi }/(R^*_{S_\phi })^{d^2}\)  is finite and effectively bounded due to [4]}, allowing us to replace \(K\)  by the finite extension \(L\)  generated by the set of values
\(\lbrace \beta \in \overline{K} : \beta ^{d^2} \in R^*_{S_\phi } \rbrace ,\) 
which depends only on \(K,d\)  and \(S\) .
If \(d\ge 3\)  or \(d=2\) , we consider respectively the curves
\(\phi (X)=Y^{d^2} \text{ or }\phi ^2(X)=Y^{d^2}.\) 
In both cases, we are able to apply [5]} to obtain a set \(T_1\)  as in the statement with the desired finiteness properties.
